New Construction of Strongly Relatively Nonexpansive Sequences by Firmly Nonexpansive-Like Mappings

Abstract

In recent works, many authors generated strongly relatively nonexpansive sequences of mappings by the sequences of firmly nonexpansive-like mappings. In this paper, we introduce a new method for construction of strongly relatively nonexpansive sequences from firmly nonexpansive-like mappings.

1. Introduction and Preliminaries

The class of firmly nonexpansive-like mappings has been introduced in [1]. Fixed point theory for such mappings can be applied to several nonlinear problems such as zero point problems for monotone operators, convex feasibility problems, convex minimization problems, equilibrium problems (see, [1,2,3,4,5] for more details).

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space X, J be a normalized duality mapping from X into dual $X^{*}$, and $S,T:C\to X$ are firmly nonexpansive-like mappings. The set of all fixed points of T is denoted by $F\left(T\right)$. It is known that if C is a bounded subset, then $F\left(T\right)$ is nonempty ([1], Theorem 7.4). We investigate asymptotic behavior of the following sequence $\left\{x_n\right\}$ in a uniformly smooth and 2-uniformly convex Banach space X.

$$x_{n+1}=Q_C{J}^{-1}(JT{x}_n-{\left({\mu }_X\right)}^{-2}J(x_n-S{x}_n))$$

(1)

for all $n\in \mathbb{N}$, where $x_1\in C$, ${\mu }_X$ denotes the uniform convexity constant of $X,$ and $Q_C$ denotes the generalized projection of X onto C. If X is a Hilbert space, then (1) is reduced to

$$x_{n+1}=T{x}_n,\mathrm{for} \mathrm{all}n\in \mathbb{N}.$$

(2)

Throughout the present paper, we denote by $\mathbb{N}$ the set of all positive integers, $\mathbb{R}$ the set of all real numbers, X a real Banach space with dual $X^{*}$, $\parallel .\parallel$ the norms of X and $X^{*}$, $\langle x,x^{*}\rangle$ the value of $x^{*}\in X^{*}$ at $x\in X$, $x_n⟶x$ strong convergence of a sequence $\left\{x_n\right\}$ of X to $x\in X$, $x_n\rightharpoonup x$ weak convergence of a sequence $\left\{x_n\right\}$ of X to $x\in X$, $S_X$ the unit sphere of X, and $B_X$ the closed unit ball of X.

Now, we present some definitions which are needed in the sequel. The normalized duality mapping of X into $X^{*}$ is defined by

$$Jx=\{x^{*}\in X^{*}:\langle x,x^{*}\rangle =\parallel x{\parallel }^2=\parallel x^{*}{\parallel }^2\}$$

(3)

for all $x\in X$. The space X is said to be smooth if

$$\underset{t\to 0}{lim}{\displaystyle \frac{\parallel x+ty\parallel -\parallel x\parallel }{t}}$$

(4)

exists for all $x,y\in S_X$. The space X is said to be uniformly smooth, if (4) converges uniformly in $x,y\in S_X$. It is said to be strictly convex, if $\parallel {\displaystyle \frac{x+y}{2}}\parallel <1$ whenever $x,y\in S_X$ and $x\ne y$. It is said to be uniformly convex, if ${\delta }_X\left(\epsilon \right)>0$ for all $\epsilon \in (0,2]$, where ${\delta }_X$ is the modulus of convexity of X defined by

$${\delta }_X\left(\epsilon \right)=inf\left\{1-\parallel {\displaystyle \frac{x+y}{2}}\parallel :x,y\in B_X,\parallel x-y\parallel ⩾\epsilon \right\}$$

(5)

for all $\epsilon \in [0,2]$.

The space X is said to be 2-uniformly convex, if there exists $c>0$ such that ${\delta }_X\left(\epsilon \right)⩾c{\epsilon }^2$ for all $\epsilon \in [0,2]$.

It is obvious that every 2-uniformly convex Banach space is uniformly convex. It is known that all Hilbert spaces are uniformly smooth and 2-uniformly convex. It is also known that all the Lebesgue spaces $L_p$ are uniformly smooth and 2-uniformly convex whenever $1<p⩽2$.

For a smooth Banach space, J is said to be weakly sequentially continuous if $\left\{J{x}_n\right\}$ converges weak to $Jx$, whenever $\left\{x_n\right\}$ is a sequence of X such that $x_n\rightharpoonup x\in X$.

Define $\phi :X\times X\to \mathbb{R}$ by

$$\phi (x,y)={\parallel x\parallel }^2-2\langle x,Jy\rangle +{\parallel y\parallel }^2$$

(6)

for all $x,y\in X$. It is known that

$$\phi (x,y)=\phi (x,z)+\phi (z,y)+2\langle x-z,Jz-Jy\rangle$$

(7)

for all $x,y,z\in X$.

Definition 1

([3]). The metric projection $P_C$ from X onto C and the generalized projection $Q_C$ from X onto C are defined by

$$P_{C}x=\underset{y\in C}{argmin}\parallel y-x\parallel ,Q_{C}x=\underset{y\in C}{argmin}\phi (y,x)$$

(8)

for all $x\in X$, respectively.

Obviously, for $x\in X$ and $z\in C$,

$$z=P_{C}x⟺\langle y-z,J(x-z)\rangle ,(\forall y\in C).$$

(9)

Also, for $x\in X$ and $z\in C$,

$$z=Q_{C}x⟺\langle y-z,Jx-Jz\rangle ,(\forall y\in C).$$

(10)

Definition 2

([1]). A mapping $T:C⟶X$ is said to be a firmly nonexpansive-like mapping, if

$$\langle Tx-Ty,J(x-Tx)-J(y-Ty)\rangle ⩾0$$

(11)

for all $x,y\in C$.

Definition 3

([1]). Let $T:C⟶X$ be a mapping. A point $p\in C$ is said to be an asymptotic fixed point of $T,$ if there exists a sequence $\left\{x_n\right\}$ of C such that $x_n\rightharpoonup p$ and $x_n-T{x}_n⟶0$. The set of all asymptotic fixed points of T is denoted by $\widehat{F}\left(T\right)$.

Definition 4

([1]). The mapping T is said to be of type $\left(r\right),$ if $F\left(T\right)$ is nonempty and $\phi (u,Tx)⩽\phi (u,x)$ for all $u\in F\left(T\right)$ and $x\in C$.

It is known that if T is a mapping of type $\left(r\right)$, then $F\left(T\right)$ is closed and convex.

Definition 5

([4]). The mapping T is said to be of type $\left(sr\right),$ if T is of type $\left(r\right)$ and $\phi (T{z}_n,z_n)⟶0$, whenever $\left\{z_n\right\}$ is a bounded sequence of C such that $\phi (u,z_n)-\phi (u,T{z}_n)⟶0$ for some $u\in F\left(T\right)$.

Definition 6

([4]). The sequence $\left\{T_n\right\}$ is said to satisfy the condition $\left(Z\right),$ if every weak subsequential limit of $\left\{x_n\right\}$ belongs to $F\left(\left\{T_n\right\}\right)$, whenever $\left\{x_n\right\}$ is a bounded sequence of C such that $x_n-T_n{x}_n⟶0$.

Now, we give some results which will be used in our main results.

Theorem 1

([5]). The space X is 2-uniformly convex if and only if there exists $\mu ⩾0$ such that

$${\displaystyle \frac{{\parallel x+y\parallel }^2+{\parallel x-y\parallel }^2}{2}}⩾{\parallel x\parallel }^2+{\parallel {\mu }^{-1}y\parallel }^2,for allx,y\in X.$$

(12)

Lemma 1

([4], Lemma 2.2). Suppose that X is 2-uniformly convex. Then

$${\left({{\displaystyle \frac{1}{\mu }}}_X\parallel x-y\parallel \right)}^2⩽\phi (x,y),for allx,y\in X.$$

(13)

Lemma 2

([1]). If $T:C⟶X$ is a firmly nonexpansive-like mapping, then $F\left(T\right)$ is a closed convex subset of X and $\widehat{F}\left(T\right)=F\left(T\right)$.

Lemma 3

([4]). Suppose that X is uniformly convex. If $S:X⟶X$ and $T:C⟶X$ are mappings of type $\left(r\right)$ such that $F\left(S\right)\cap F\left(T\right)$ is nonempty and S or T is of type $\left(sr\right)$, then $ST:C⟶X$ is of type $\left(r\right)$ and $F\left(ST\right)=F\left(S\right)\cap F\left(T\right)$. Further, if both S and T are of type $\left(sr\right)$, then so is $ST$.

Lemma 4

([4]). Suppose that X is uniformly convex. Let $\left\{S_n\right\}$ be a sequence of mappings of X into itself and $\left\{T_n\right\}$ a sequence of mappings of C into X such that $F\left(\left\{S_n\right\}\right)\cap F\left(\left\{T_n\right\}\right)$ is nonempty, both $\left\{S_n\right\}$ and $\left\{T_n\right\}$ are of type $\left(sr\right),$ and $S_n$ or $T_n$ is of type $\left(sr\right)$ for all $n\in \mathbb{N}$. Then the following holds:

(i)

$\left\{S_n{T}_n\right\}$ is of type $\left(sr\right)$;

(ii)

if X is uniformly smooth and both $\left\{S_n\right\}$ and $\left\{T_n\right\}$ satisfy the condition $\left(Z\right)$, then so does $\left\{S_n{T}_n\right\}$.

Theorem 2

([4]). Let X be a smooth and uniformly convex Banach space, C a nonempty closed convex subset of X, and $\left\{T_n\right\}$ a sequence of mappings of C into X such that $\left\{T_n\right\}$ is of type $\left(sr\right)$ and $\left\{T_n\right\}$ satisfies the condition $\left(Z\right)$. If $T_n\left(C\right)\subset C$ for all $n\in \mathbb{N}$ and J is weakly sequentially continuous, then the sequence $\left\{x_n\right\}$ defined by $x_1\in C$ and $x_{n+1}=T_n{x}_n$ for all $n\in \mathbb{N}$ converges weakly to the strong limit of $\left\{Q_F{x}_n\right\}$.

Now, we construct a new strongly relatively nonexpansive sequence from a given sequence of firmly nonexpansive-like mappings with a common fixed point in Banach spaces.

2. Main Results

The following results will be used in the sequel of the paper.

Lemma 5.

Let C be a nonempty closed convex subset of a smooth, strictly convex, 2-uniformly convex and reflexive Banach space X. Suppose that $(S,T)$ is a pair of firmly nonexpansive-like mappings of C into X and let $F=F\left(S\right)\cap F\left(T\right)\ne \varnothing$. Let U be a mapping of C into X defined by $U=J^{-1}(JT-\beta J(I-S))$, where $\beta >0$ and I denotes the identity mapping on C. Then

$$\phi (u,Ux)+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{2}{{\mu }_X^2}}-\beta \right){\parallel Ux-Tx\parallel }^2⩽\phi (u,Tx)$$

for all $u\in F\left(U\right)$ and $x\in C$.

Proof. 

Let $u\in F\left(U\right)$ and $x\in C$ be given. Then, from (7) and the definition of $U,$ it follows that

$$\begin{array}{cc}\hfill \phi (u,Ux)+\phi (Ux,Tx)-\phi (u,Tx)& =2\langle u-Ux,JTx-JUx\rangle \hfill \\ \hfill & =2\beta \langle u-Ux,J(x-Sx)\rangle .\hfill \end{array}$$

(14)

Since S is firmly nonexpansive-like and $u\in F\left(S\right)$, we know that

$$\begin{array}{cc}\hfill \langle u-Ux,J(x-Sx)\rangle & =\langle u-Sx,J(x-Sx)\rangle +\langle Sx-Ux,J(x-Sx)\rangle \hfill \\ \hfill & =\langle Sx-Ux,J(x-Sx)\rangle .\hfill \end{array}$$

(15)

On the other hand, we have

$$\begin{array}{cc}\hfill \langle Sx-Ux,J(x-Sx)\rangle & ={-\parallel Sx-Tx\parallel }^2+\langle Tx-Ux,J(x-Sx)\rangle \hfill \\ \hfill & ⩽-(\parallel Sx-Tx{\parallel }^2-\parallel Tx-Ux\parallel \parallel x-Sx\parallel )\hfill \\ \hfill & ⩽{-(\parallel Sx-x\parallel }^2-{\displaystyle \frac{1}{2}}{\parallel Ux-Tx\parallel )}^2+{\displaystyle \frac{1}{4}}{\parallel Ux-Tx\parallel }^2\hfill \\ \hfill & ⩽{\parallel Ux-Tx\parallel }^2.\hfill \end{array}$$

(16)

Since $\beta >0$, from (14)–(16), we deduce that

$$\begin{array}{c}\hfill \phi (u,Ux)+\phi (Ux,Tx)-\phi (u,Tx)⩽{2\beta \parallel Ux-Tx\parallel }^2.\end{array}$$

(17)

Since X is 2-uniformly convex, Lemma 1 implies that

$$\begin{array}{c}\hfill {\left({\mu }_X\right)}^{-2}{\parallel Ux-Tx\parallel }^2⩽\phi (Ux,Tx).\end{array}$$

(18)

By (17) and (18), we obtain the desired inequality. □

Now, we present the construction of strongly relatively nonexpansive sequences in the following.

Theorem 3.

Let C be a nonempty closed convex subset of a smooth and 2-uniformly convex Banach space X;

(i)

$\left\{T_n\right\}$, $\left\{S_n\right\}$ are sequences of firmly nonexpansive-like mappings from C into X such that $F=F\left(\left\{T_n\right\}\right)\cap F\left(\left\{S_n\right\}\right)$ is nonempty;

(ii)

$\left\{U_n\right\}$ is a sequence of mappings from C into X defined by

$$U_n=J^{-1}\left(J{T}_n-{\beta }_{n}J(I-S_n)\right)$$

for all $n\in \mathbb{N}$, where ${\beta }_n$ is a sequence of real numbers such that $0<{inf}_n{\beta }_n$ and ${sup}_n{\beta }_n<2{\left({\mu }_X\right)}^{-2}$ and I denotes the identity mapping on C.

Then $F\left(\left\{U_n\right\}\right)\subset F\left(\left\{S_n\right\}\right)\cap F\left(\left\{T_n\right\}\right)$ and $\left\{U_n\right\}$ is of type $\left(sr\right)$. Also, if X is uniformly smooth and $\left\{S_n\right\}$ satisfies the condition $\left(Z\right)$, then $\left\{U_n\right\}$ satisfies the condition $\left(Z\right)$.

Proof. 

We can easily see that $F\left(\left\{U_n\right\}\right)\subset F\left(\left\{S_n\right\}\right)\cap F\left(\left\{T_n\right\}\right)$. At first, we show that $\left\{U_n\right\}$ is of type $\left(sr\right)$.

Note that $F\left(\left\{U_n\right\}\right)$ is nonempty. By Lemma 5, we also know that each $U_n$ is a mapping of type $\left(r\right)$ from C into X.

Suppose that $\left\{T_n{z}_n\right\}$ is a bounded sequence of C such that

$$\phi (u,T_n{z}_n)-\phi (u,U_n{T}_n{z}_n)⟶0$$

for some $u\in F\left(\left\{U_n\right\}\right)$. Then, it follows from Lemma 5 that

$$\begin{array}{c}\hfill 0⩽{\displaystyle \frac{1}{2}}({\displaystyle \frac{2}{{\mu }_X^2}}-{\beta }_n){\parallel U_n{z}_n-T_n{z}_n\parallel }^2⩽\phi (u,T_n{z}_n)-\phi (u,U_n{z}_n).\end{array}$$

(19)

Thus, it follows from ${sup}_n{\beta }_n<2{\left({\mu }_X\right)}^{-2}$ that $\parallel U_n{z}_n-T_n{z}_n\parallel ⟶0$. Consequently, we have $\phi (U_n{z}_n,T_n{z}_n)⟶0$ and hence $\left\{U_n\right\}$ is of type $\left(sr\right)$. Now, we present the proof of part $\left(ii\right)$. Suppose that X is uniformly smooth and $\left\{S_n\right\}$ satisfies the condition $\left(Z\right)$. Let p be a weak subsequential limit of a bounded sequence $\left\{x_n\right\}$ of C such that $T_n{x}_n-U_n{x}_n⟶0$. By the definition of $U_n$, we have

$$\begin{array}{c}\hfill J(x_n-S_n{x}_n)={\displaystyle \frac{1}{{\beta }_n}}(J{T}_n{x}_n-J{U}_n{x}_n)\end{array}$$

(20)

for all $n\in \mathbb{N}$. Since J is uniformly norm-to-norm continuous on each nonempty bounded subset of X and ${sup}_n{\displaystyle \frac{1}{{\beta }_n}}<\infty$, it follows from (20) that

$$\parallel x_n-S_n{x}_n\parallel ={\displaystyle \frac{1}{{\beta }_n}}\parallel J{T}_n{x}_n-J{U}_n{x}_n\parallel ⟶0.$$

From our assumptions, we know that $p\in F\supset F\left(\left\{U_n\right\}\right)$. Therefore, $\left\{U_n\right\}$ satisfies the condition $\left(Z\right)$. □

Remark 1.

It is notable that every nonexpansive mapping T is a mapping of type $\left(r\right)$, but the converse is not necessarily satisfied in a Hilbert space. For instance, let $T:\mathbb{R}⟶\mathbb{R}$ be defined by $Tx=x^2$, then T is of type $\left(r\right)$ and is neither nonexpansive nor of type $\left(sr\right)$. Also, let $T:{\mathbb{R}}^{+}⟶{\mathbb{R}}^{+}$ be defined by $Tx=\sqrt{x}$. Then T is a mapping of type $\left(sr\right)$.

Remark 2.

For a mapping T from C into X, the following assertions hold:

(a)

T is of type $\left(sr\right)$ if and only if $\{T,T,\cdots \}$ is of type $\left(sr\right)$;

(b)

$\widehat{F}\left(T\right)=F\left(T\right)$ if and only if $\{T,T,\cdots \}$ satisfies the condition $\left(Z\right)$.

Corollary 1.

Let $(S,T)$ be a pair of firmly nonexpansive-like mappings from C into X such that $F\left(T\right)\cap F\left(S\right)$ are nonempty and U be a mapping from C into X which is defined by

$$U=J^{-1}(JT-\beta J(I-S))$$

where $0<\beta <2{\left({\mu }_X\right)}^{-2}$. Then the following assertions hold:

(i)

$F\left(U\right)\subset F\left(T\right)\cap F\left(S\right)$ and U is of type $\left(sr\right)$;

(ii)

if X is uniformly smooth, then $\widehat{F}\left(U\right)=F\left(U\right)$.

Theorem 4.

Let $\left\{V_n\right\}$ be a sequence of mappings from C into itself which are defined by

$$V_n=Q_C{U}_n$$

for all $n\in \mathbb{N}$. Then the following consequences hold:

(i)

$F\left(\left\{V_n\right\}\right)\subset F$ and $\left\{V_n\right\}$ is of type $\left(sr\right)$;

(ii)

if X is uniformly smooth and $\left\{S_n\right\}$ satisfies the condition $\left(Z\right)$, then so does $\left\{V_n\right\}$.

Proof. 

We know that $F\left(V_n\right)\subset F\left(T_n\right)\cap F\left(S_n\right)$ for all $n\in N$ and hence $F\left(\left\{V_n\right\}\right)\subset F\ne \varnothing$. We first show that $\left\{V_n\right\}$ is of type $\left(sr\right)$. From $\left(i\right)$ of Corollary 1, we know that each $U_n$ is of type $\left(sr\right)$. Since $Q_C$ is of type $\left(sr\right)$ from X into itself and

$$F\left(Q_C\right)\cap F\left(U_n\right)\subset F\left(T_n\right)\cap F\left(S_n\right)\supset F\ne \varnothing ,$$

Lemma 3 implies that each $V_n=Q_C{U}_n$ is also of type $\left(sr\right)$.

Since $\{Q_C,Q_C,...\}$ is of type $\left(sr\right)$ by Remark 2, $\left\{U_n\right\}$ is of type $\left(sr\right)$ by Theorem 3, and

$$F\left(Q_C\right)\cap F\left(\left\{U_n\right\}\right)\subset F\ne \varnothing ,$$

the part $\left(i\right)$ of Lemma 4 implies that $\left\{V_n\right\}$ is of type $\left(sr\right)$.

We finally show the part $\left(ii\right)$. Suppose that X is uniformly smooth and $\left\{S_n\right\}$ satisfies the condition $\left(Z\right)$. Since C is weakly closed, we can easily see that $\widehat{F}\left(Q_C\right)=F\left(Q_C\right)=C$. This implies that $\{Q_C,Q_C,...\}$ satisfies the condition $\left(Z\right)$. From Theorem 3, we know that $\left\{U_n\right\}$ satisfies the condition $\left(Z\right)$. Thus, the part $\left(ii\right)$ of Lemma 4 implies the conclusion. □

As a direct consequence of Theorems 2 and 4, we obtain the following result.

Theorem 5.

Let X be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of X, $\left\{T_n\right\}and\left\{S_n\right\}$ be two sequences of firmly nonexpansive-like mappings from C into X such that $F=F\left(\left\{T_n\right\}\right)\cap F\left(\left\{S_n\right\}\right)$ is nonempty and $\left\{S_n\right\}$ satisfies the condition $\left(Z\right)$, ${\beta }_n$ be a sequence of real numbers such that

$$0<\underset{n}{inf}{\beta }_n,\underset{n}{sup}{\beta }_n<2{\left({\mu }_X\right)}^{-2},$$

and $\left\{x_n\right\}$ be a sequence defined by $x_1\in C$ and

$$x_{n+1}=Q_C{J}^{-1}\left(J{T}_n{x}_n-{\beta }_{n}J(x_n-S_n{x}_n)\right)$$

for all $n\in \mathbb{N}$. If J is weakly sequentially continuous, then $\left\{x_n\right\}$ converges weakly to the strong limit of $\left\{Q_F{x}_n\right\}$.